This periodically updated reference resource is intended to put eager researchers on the path to fame and (perhaps) fortune.

Abstract. Aim: To present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Ascend from Nim to chess in small strides at a gradient that’s not too steep. Presentation: Informal; examples of games sampled from various strategic viewing points along scenic mountain trails illustrate the theory. 1.

Abstract Studying the precise nature of the complexity of games enables gamesters to attain a deeper understanding of the difficulties involved in certain new and old open game problems, which is a key to their solution. For algorithmicians, such studies provide new interesting algorithmic challenges. Substantiations of these claims are illustrated on hand of many sample games, leading to a definition of the tractability, polynomiality and efficiency of subsets of games. In particular, there are tractable games that are not polynomial, polynomial games that are not efficient. We also define and explore the nature of the subclasses PlayGames and MathGames.

Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a well-stirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and

A “meta ” (pseudo-) algorithm is described that, for any fixed k, finds a fast (O�log�a��) algorithm for playing 3-rowed Chomp, starting with the first, second, and third rows of lengths a, b, and c, respectively, where c ≤ k, buta and b are arbitrary. © 2001 Academic Press How to Play CHOMP David Gale’s famous game of Chomp [Ch] starts out with an M by N chocolate bar, in which the leftmost-topmost square is poisonous. Players take turns picking squares. In his or her (or its) turn, a player must pick one of the remaining squares, and eat it along with all the squares that are “to its right and below it. ” Using matrix-notation with the poisonous square being entry (1, 1), and the initial position consisting of the whole bar ��i � j��1 ≤ i ≤ M � 1 ≤ j ≤ N�, then picking the square �i0�j0 � means that one has to eat all the remaining squares �i � j � for which both i ≥ i0 and j ≥ j0 hold. The player that eats the poisonous (leftmost-topmost) square loses. Of course picking (1, 1) kills you, so a non-suicidal player will not play that move unless forced to. For example, if M = 4 and N = 3, then the initial game-position is

Aim: Present a systematic development of part of the theory of combinatorial games from the ground up. Approach: Computational complexity. Combinatorial games are completely determined; the questions of interest are efficiencies of strategies. Methodology: Divide and conquer. Isolate the various difficulties separating hard from easy games, and attack them individually. Presentation: Informal; examples of games sampled from various levels illustrate the theory, with emphasis on formulating and motivating new and old research problems.

Two players I and II are playing the Hackendot game, a slight variant of Hackenbush (see Berlekamp, Conway and Guy [1]) played on a partially ordered set P : First I chooses x 2 P and deletes the final section generated by x: he leaves the order P x , then II chooses y 2 P x and deletes the final section generated by y... and so on alternately. The player who cannot move loses the game. This kind of game was introduced by Von-Neumann when the order P is a reverse ordered tree (the root is maximal). He concluded in this case that player I wins by a strategy stealing argument. In [5], ' Ulehla described how player I could win. We give an easy argument to calculate such a strategy for a wider class of partially ordered sets.

pen Problems #10 (Spring 1993). Albertson had asked for the minimum value of the independence ratio ff(G)=n(G) for trianglefree planar graphs; in particular, does it exceed 1/3? Since-triangle free planar graphs are 3-colorable (Grotzsch's Theorem), the value is at least 1/3. Tovey and Steinberg have proved that ff(G)=n(G) ? 1=3 for every triangle-free graph. This is best possible: Fraughnaugh [9a] published a sequence of triangle-free planar graphs such that the kth graph has 3k + 2 vertices and has independence number k + 1. Steinberg [26] has published a survey article on problems related to Grotzsch's Theorem, and Carsten Thomassen [28] has recently found a short proof of a strengthening of Grotzsch's Theorem. Aviezri Fraenkel of the Weizmann Institute offers further insight

We show that it is extremely difficult to devise incentive schemes that distinguish between fund managers who cannot deliver excess returns from those who can, unless investors have specific knowledge of the investment strategies being employed. Using a ‘performance‐mimicking ’ argument, we show that any fee structure that does not assess penalties for underperformance can be gamed by unskilled managers to generate fees that are at least as high, per dollar of expected returns, as the fees of the most skilled managers. We show further that standard proposals to reform the fee structure, such as imposing high water marks, delaying managers ’ bonus payments, forcing them to hold an equity stake, or assessing penalties for underperformance, are not enough to separate the skilled from the unskilled. We conclude that skilled managers will have to find ways other than their track records to distinguish themselves from the unskilled, or else the latter may drive out the former as in a classic lemons market.

In this article, dedicated with admiration and friendship to chaos and difference (and hence recurrence) equations guru Saber Elaydi, I give a new approach and a new algorithm for Chomp, David Gale’s celebrated combinatorial game. This work is inspired by Xinyu Sun’s “ultimate-periodicity ” conjecture and by its brilliant proof by highschool student Steven Byrnes. The algorithm is implemented in a Maple package BYRNES accompanying this article. By looking at the output, and inspired by previous work of Andries Brouwer, I speculate that Chomp is chaotic, in a yet-to-be-made-precise sense, because the losing positions are given by “weird ” recurrences.